Advanced High-School Mathematics

288 CHAPTER 5 Series and Differential Equations

2. Determine the interval convergence of each of the power series
   below:

(a)

∑∞

n=0

nxn

n+ 1

(b)

∑∞

n=0

nxn

2 n

(c)

∑∞

n=1

xn

n 2 n

(d)

∑∞

n=0

(−1)n(x−2)^2 n

3 n

(e)

∑∞

n=1

(−1)n(x+ 2)n

n 2 n

(f)

∑∞

n=0

(−1)n(x+ 2)n

2 n

(g)

∑∞

n=0

(2x)n

n!

(h)

∑∞

n=2

(−1)nxn

nlnn 2 n

(i)

∑∞

n=1

(−1)nnlnnxn

2 n

(j)

∑∞

n=0

(3x−2)n

2 n

5.4 Polynomial Approximations; Maclaurin and Tay-

lor Expansions

Way back in our study of thelinearization of a functionwe saw that
it was occassionally convenient and useful to approximate a function by
one of its tangent lines. More precisely, iff is a differentiable function,
and if a is a value in its domain, then we have the approximation

f′(a)≈

f(x)−f(a)

x−a

, which results in

f(x) ≈ f(a) +f′(a)(x−a) forxneara.

A graph of this situation should help remind the student of how good
(or bad) such an approximation might be: